Theory and Definitions of Classical Orthogonal Polynomials.

\( \def\ud{{\rm\,d}} \def\ii{{\rm i}} \def\fl{{\rm\,fl}} \def\abs#1{\left|{#1}\right|} \def\norm#1{\left\|{#1}\right\|} \def\conj#1{\overline{#1}} \)

Classical Orthogonal Polynomials

We begin by reviewing standard notation that is used to discuss classical orthogonal polynomials. In particular, \(L^2(D,\ud\mu)\) will be the Hilbert space of square integrable functions defined on \(D\subset\mathbb{R}^d\) with positive Borel measure \(\ud\mu\). Classical orthogonal polynomials are all defined on a subset of the real line:

• Jacobi polynomials, \(P_n^{(\alpha,\beta)}(x)\), are orthogonal polynomials with respect to \(L^2([-1,1], (1-x)^\alpha(1+x)^\beta\ud x)\);
• Hermite polynomials, \(H_n(x)\), are orthogonal polynomials with respect to \(L^2(\mathbb{R}, e^{-x^2}\ud x)\); and,
• generalized Laguerre polynomials, \(L_n^{(\alpha)}(x)\), are orthogonal polynomials with respect to \(L^2(\mathbb{R}^+, x^\alpha e^{-x}\ud x)\).

We use the standard normalizations defined by DLMF. When an orthogonal polynomial appears with a tilde overtop, this implies orthonormalization.

Let \(\{\phi_n(x)\}_{n\in\mathbb{N}_0}\) be a family of orthogonal functions with respect to \(L^2(\hat{D},\ud\hat{\mu})\) and let \(\{\psi_n(x)\}_{n\in\mathbb{N}_0}\) be another family of orthogonal functions with respect to \(L^2(D,\ud\mu)\). The connection coefficients:

\[ c_{\ell,n} = \frac{\langle \psi_\ell, \phi_n\rangle_{\ud\mu}}{\langle \psi_\ell, \psi_\ell\rangle_{\ud\mu}}, \]

allow for the expansion:

\[ \phi_n(x) = \sum_{\ell=0}^\infty c_{\ell,n} \psi_\ell(x). \]

The fast and backward stable transforms in this library are based on data-sparse factorizations of the connection problems. That is, rather than using recurrence relations to evaluate orthogonal polynomials, we convert their expansions using sequences of Givens rotations. This procedure is more accurate because Givens rotations are \(2\)-norm preserving, while recurrence relations inject new energy at each step. The Givens rotation solutions to the connection problem may be distilled to three results that relate different weighted orthonormal polynomials with respect to the same Hilbert space.

Let \(I_{m\times n}\) denote the rectangular identity matrix with ones on the main diagonal and zeros everywhere else.

The Jacobi connection problem

Let \(G_n\) denote the Givens rotation:

\[ G_n = \begin{pmatrix} 1 & \cdots & 0 & 0 & \cdots & 0\\ \vdots & \ddots & \vdots & \vdots & & \vdots\\ 0 & \cdots & c_n & s_n & \cdots & 0\\ 0 & \cdots & -s_n & c_n & \cdots & 0\\ \vdots & & \vdots & \vdots & \ddots & \vdots\\ 0 & \cdots & 0 & 0 & \cdots & 1\\ \end{pmatrix}, \]

where the sines \(s_n = \sin\theta_n\) and the cosines \(c_n = \cos\theta_n\), for some \(\theta_n\in[0,2\pi)\), are in the intersections of the \(n^{\rm th}\) and \(n+1^{\rm st}\) rows and columns, embedded in the identity of a conformable size.

The connection coefficients between \((1-x)\tilde{P}_n^{(\alpha+2,\beta)}(x)\) and \(\tilde{P}_{\ell}^{(\alpha,\beta)}(x)\) are:

\[ c_{\ell,n}^{(\alpha,\beta)} = \left\{\begin{array}{ccc} (\alpha+1)\sqrt{\frac{(2\ell+\alpha+\beta+1)\Gamma(\ell+\alpha+\beta+1)\Gamma(\ell+\alpha+1)}{\Gamma(\ell+\beta+1)\Gamma(\ell+1)}\frac{(2n+\alpha+\beta+3)\Gamma(n+\beta+1)\Gamma(n+1)}{\Gamma(n+\alpha+\beta+3)\Gamma(n+\alpha+3)}}, & {\rm for} & \ell \le n,\\ -\sqrt{\frac{(n+1)(n+\beta+1)}{(n+\alpha+2)(n+\alpha+\beta+2)}}, & {\rm for} & \ell = n+1,\\ 0, & & {\rm otherwise}. \end{array} \right. \]

Furthermore, the matrix of connection coefficients \(C^{(\alpha,\beta)} \in \mathbb{R}^{(n+2)\times (n+1)}\) may be represented via the product of \(n+1\) Givens rotations:

\[ C^{(\alpha,\beta)} = G_0^{(\alpha,\beta)}G_1^{(\alpha,\beta)}\cdots G_{n-1}^{(\alpha,\beta)}G_n^{(\alpha,\beta)} I_{(n+2)\times (n+1)}, \]

where the sines and cosines for the Givens rotations are given by:

\[ s_n^{(\alpha,\beta)} = \sqrt{\frac{(n+1)(n+\beta+1)}{(n+\alpha+2)(n+\alpha+\beta+2)}},\quad{\rm and}\quad c_n^{(\alpha,\beta)} = \sqrt{\frac{(\alpha+1)(2n+\alpha+\beta+3)}{(n+\alpha+2)(n+\alpha+\beta+2)}}. \]

Since \(P_n^{(\alpha,\beta)}(-x) = (-1)^n P_n^{(\beta,\alpha)}(x)\), the connection coefficients between \((1+x)\tilde{P}_n^{(\alpha,\beta+2)}(x)\) and \(\tilde{P}_{\ell}^{(\alpha,\beta)}(x)\) are easily obtained by reversing the roles of \(\alpha\) and \(\beta\) above and multiplying each coefficient by \((-1)^{n-\ell}\). For the Givens rotations, this negates the sines.

The generalized Laguerre connection problem

Let \(G_n\) denote the real Givens rotation as above.

The connection coefficients between \(x\tilde{L}_n^{(\alpha+2)}(x)\) and \(\tilde{L}_{\ell}^{(\alpha)}(x)\) are:

\[ c_{\ell,n}^{(\alpha)} = \left\{\begin{array}{ccc} (\alpha+1)\sqrt{\frac{\Gamma(\ell+\alpha+1)}{\Gamma(\ell+1)}\frac{\Gamma(n+1)}{\Gamma(n+\alpha+3)}}, & {\rm for} & \ell \le n,\\ -\sqrt{\frac{n+1}{n+\alpha+2}}, & {\rm for} & \ell = n+1,\\ 0, & & {\rm otherwise}. \end{array} \right. \]

Furthermore, the matrix of connection coefficients \(C^{(\alpha)} \in \mathbb{R}^{(n+2)\times (n+1)}\) may be represented via the product of \(n+1\) Givens rotations:

\[ C^{(\alpha)} = G_0^{(\alpha)}G_1^{(\alpha)}\cdots G_{n-1}^{(\alpha)}G_n^{(\alpha)} I_{(n+2)\times (n+1)}, \]

where the sines and cosines for the Givens rotations are given by:

\[ s_n^{(\alpha)} = \sqrt{\frac{n+1}{n+\alpha+2}},\quad{\rm and}\quad c_n^{(\alpha)} = \sqrt{\frac{\alpha+1}{n+\alpha+2}}. \]

The ultraspherical connection problem

Ultraspherical polynomials are proportional to Jacobi polynomials with the parameters equal, say \(\beta=\alpha\). The connection coefficients respect their even-odd symmetry and this is reflected in the Givens rotations as well.

Let \(G_n\) denote the real Givens rotation:

\[ G_n = \begin{pmatrix} 1 & \cdots & 0 & 0 & 0 & \cdots & 0\\ \vdots & \ddots & \vdots & \vdots & \vdots & & \vdots\\ 0 & \cdots & c_n & 0 & s_n & \cdots & 0\\ 0& \cdots & 0 & 1 & 0 & \cdots & 0\\ 0 & \cdots & -s_n & 0 & c_n & \cdots & 0\\ \vdots & & \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & \cdots & 0 & 0 & 0 & \cdots & 1\\ \end{pmatrix}, \]

where the sines \(s_n = \sin\theta_n\) and the cosines \(c_n = \cos\theta_n\), for some \(\theta_n\in[0,2\pi)\), are in the intersections of the \(n^{\rm th}\) and \(n+2^{\rm nd}\) rows and columns, embedded in the identity of a conformable size.

The connection coefficients between \((1-x^2)\tilde{P}_n^{(\alpha+2,\alpha+2)}(x)\) and \(\tilde{P}_\ell^{(\alpha,\alpha)}(x)\) are:

\[ c_{\ell,n}^{(\alpha)} = \left\{\begin{array}{ccc} (2\alpha+2)\sqrt{\frac{(2\ell+2\alpha+1)\Gamma(\ell+2\alpha+1)}{\Gamma(\ell+1)}\frac{(2n+2\alpha+5)\Gamma(n+1)}{\Gamma(n+2\alpha+5)}}, & {\rm for} & \ell \le n,\quad \ell+n\hbox{ even},\\ -\sqrt{\frac{(n+1)(n+2)}{(n+2\alpha+3)(n+2\alpha+4)}}, & {\rm for} & \ell = n+2,\\ 0, & & {\rm otherwise}. \end{array} \right. \]

Furthermore, the matrix of connection coefficients \(C^{(\alpha)} \in \mathbb{R}^{(n+3)\times (n+1)}\) may be represented via the product of \(n+1\) Givens rotations:

\[ C^{(\alpha)} = G_0^{(\alpha)}G_1^{(\alpha)}\cdots G_{n-1}^{(\alpha)}G_n^{(\alpha)} I_{(n+3)\times (n+1)}, \]

where the sines and cosines for the Givens rotations are given by:

\[ s_n^{(\alpha)} = \sqrt{\frac{(n+1)(n+2)}{(n+2\alpha+3)(n+2\alpha+4)}},\quad{\rm and}\quad c_n^{(\alpha)} = \sqrt{\frac{(2\alpha+2)(2n+2\alpha+5)}{(n+2\alpha+3)(n+2\alpha+4)}}. \]